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Semilinear integro-differential equations with compact semigroups. (English) Zbl 0919.34055
The paper is concerned with the integro-differential equation \[ \frac{du}{dt}+Au(t)= f(t,u(t))+\int_{t_0}^ta(t-s)g(s,u(s))ds, \;0\leq t_0<T_0\leq\infty, \;u(t_0)=u_0, \tag{1} \] in a Banach space \(X\) with \(-A\) being the generator of a compact semigroup \(T(t)\), \(t\geq 0\); \(f,g: J\times U\to X\), \(J=[t_0,T)\), \(U\) is an open set in \(X\), \(a\in L^1(J)\), \(u_0\in U\). It extends known results to the case \(g\not=0\). It is proved that if the nonlinear maps \(f\) and \(g\) are continuous and \(a\) is locally integrable in \(J\), then for every \(u_0\in X\) the initial problem (1) has a local mild solution \(u(t)\) (that is \(u(t)\), \(t\in[t_0,t_1)\) is a solution to a corresponding integral equation to (1) for some \(t_1<T_0\)). For the global case it is proved that if \(f,g\) are continuous, map bounded subsets to bounded subsets and \(a\) is locally integrable, then the initial problem (1) has a mild solution \(u(t)\) on the maximal interval of existence \([t_0,T_{\max})\).

34G20 Nonlinear differential equations in abstract spaces
47G20 Integro-differential operators
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
47D06 One-parameter semigroups and linear evolution equations
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